Welcome to the World of Mathematical Proof!

In the previous chapter, you learned how to spot patterns and create general rules (like \(y = 2x + 1\)). But in the IB MYP, finding the rule is only half the battle! To score high in Criterion B: Investigating Patterns, you need to prove that your rule actually works. This process is split into two parts: Verifying and Justifying.

Think of it like being a detective: Verifying is checking your clues to see if they fit, while Justifying is explaining to the jury exactly how you solved the crime. Let’s dive in!


1. Verifying Your Rule: The "Test Drive"

Verification is the process of testing your general rule against specific cases to see if it produces the correct results. It is like "double-checking" your work.

How to Verify Step-by-Step:

1. Pick a new term: Choose a position in your pattern that you haven't used yet (for example, if you found the rule using terms 1, 2, and 3, try testing term 5).
2. Find the "Actual" value: Look at your diagram or list of numbers and manually count or calculate what the answer should be.
3. Use your "Rule" value: Plug the term number into your formula.
4. Compare: If the "Actual" value matches your "Rule" value, your rule is verified for that case!

Example:
If your rule for a sequence is \(T_n = 3n + 2\).
To verify for \(n = 4\):
Actual value (from your diagram): 14
Rule value: \(3(4) + 2 = 12 + 2 = 14\)
Since \(14 = 14\), the rule is verified!

Quick Tip: In an MYP eAssessment, always show at least two verifications with different numbers to show your rule is consistent.


2. Justifying Your Rule: The "Why"

Justification is much deeper than verification. While verification shows that the rule does work, justification explains why it works for every possible case.

Method A: Visual Justification

Often, patterns come from shapes or diagrams. You can justify your rule by breaking the diagram apart.
Example: If you have a pattern of tiles that grows by 3 each time, you might say: "The '+2' in my rule \(3n + 2\) represents the two constant tiles at the ends, and the '3n' represents the three new tiles added to the center for every step \(n\)."

Method B: Algebraic Justification

For more advanced students, you might justify a rule by showing how it was derived from other known formulas. This often involves expanding brackets or simplifying expressions to show that two different ways of looking at the pattern result in the same rule.
\(2(n + 1) + 3n \implies 2n + 2 + 3n \implies 5n + 2\)

Don't worry if this seems tricky at first! Justification is about logic. If you can explain the link between the structure of the pattern and the numbers in your formula, you are justifying.

Key Takeaway: Verification uses numbers to test; Justification uses logic or diagrams to explain.


3. Using Proper Mathematical Language (Criterion C)

To communicate your reasoning clearly, you should use specific mathematical symbols and "connective" words. This helps your examiner follow your logic.

  • Substitution: The act of replacing a letter with a number (e.g., "Substituting \(n = 10\) into the rule...").
  • Therefore (\(\therefore\)): Use this before your final conclusion.
  • Implies (\(\implies\)): Use this to show the next logical step in an equation.
  • General Rule: A formula that works for any value of \(n\).

Common Mistake to Avoid: Don't just write a pile of numbers! Write a sentence like: "Since the rule matches the results for \(n=5\) and \(n=6\), the rule is verified."


4. Summary Checklist for Success

When you are finishing an investigation task, ask yourself these three questions:

Did I verify?

Did I test my rule with at least two different values and show the math? (e.g., \(n=5\) and \(n=10\))

Did I justify?

Did I explain why the parts of my formula (like the coefficient or the constant) exist based on the pattern's structure?

Is my communication clear?

Did I use symbols like \(=\) and \(T_n\) correctly? Did I label my steps?

Did you know? In the MYP eAssessment, Task 2 is dedicated to this process. Even if you get the "wrong" rule, you can still earn high marks in Criterion B if you verify it correctly and show logical reasoning in your justification!


Quick Review

Verification: Putting numbers into your rule to see if it stays true for specific examples.
Justification: Giving a reasoned argument (using words, diagrams, or algebra) to prove the rule is true for all cases.
Criterion B: The part of your grade that looks at how well you investigate and prove patterns.